In distributed systems where messages are asynchronous and failures can be Byzantine, we have to use at least n = 3f + 1 replicas in total to tolerate f faulty replicas. But why?
Consider this scenario: a client sends the same command to each of n servers and then waits for servers to execute the command and send back the result. If we have f byzantine servers, then up to f messages could not be trusted, or up to f servers could not responding. So the client must be able to function with n - f responses. In addition, as the messages are asynchronous, these messages could be delayed for an indefinite amount of time. So if there are f messages not received by client, then it could be also be that these f messages are not from faulty servers, but instead caused by networking partition or slow responding non-faulty servers. In this case, where the f messages are from slow but non-faulty servers, we will have n - f messages remaining and out of these n - f messages, there could be at most f faulty messages again from byzantine servers, so the worst case we will have n - 2f correct message. For the system to function properly, the correct messages must out number faulty messages for client to identify which is which, which implies we should have n - 2f > f, so n > 3f where f is the number of byzantine servers.